Properties

Label 2-234-13.12-c5-0-0
Degree $2$
Conductor $234$
Sign $-0.554 + 0.832i$
Analytic cond. $37.5298$
Root an. cond. $6.12615$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s − 16·4-s + 68i·5-s − 82i·7-s − 64i·8-s − 272·10-s + 390i·11-s + (507 + 338i)13-s + 328·14-s + 256·16-s − 1.73e3·17-s + 1.07e3i·19-s − 1.08e3i·20-s − 1.56e3·22-s − 2.10e3·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 1.21i·5-s − 0.632i·7-s − 0.353i·8-s − 0.860·10-s + 0.971i·11-s + (0.832 + 0.554i)13-s + 0.447·14-s + 0.250·16-s − 1.45·17-s + 0.682i·19-s − 0.608i·20-s − 0.687·22-s − 0.829·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.554 + 0.832i$
Analytic conductor: \(37.5298\)
Root analytic conductor: \(6.12615\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :5/2),\ -0.554 + 0.832i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6027078156\)
\(L(\frac12)\) \(\approx\) \(0.6027078156\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
3 \( 1 \)
13 \( 1 + (-507 - 338i)T \)
good5 \( 1 - 68iT - 3.12e3T^{2} \)
7 \( 1 + 82iT - 1.68e4T^{2} \)
11 \( 1 - 390iT - 1.61e5T^{2} \)
17 \( 1 + 1.73e3T + 1.41e6T^{2} \)
19 \( 1 - 1.07e3iT - 2.47e6T^{2} \)
23 \( 1 + 2.10e3T + 6.43e6T^{2} \)
29 \( 1 - 1.69e3T + 2.05e7T^{2} \)
31 \( 1 - 1.43e3iT - 2.86e7T^{2} \)
37 \( 1 + 8.85e3iT - 6.93e7T^{2} \)
41 \( 1 - 6.76e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.69e4T + 1.47e8T^{2} \)
47 \( 1 + 2.51e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.82e4T + 4.18e8T^{2} \)
59 \( 1 - 2.12e4iT - 7.14e8T^{2} \)
61 \( 1 + 5.45e3T + 8.44e8T^{2} \)
67 \( 1 + 4.45e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.77e4iT - 1.80e9T^{2} \)
73 \( 1 + 3.10e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.53e4T + 3.07e9T^{2} \)
83 \( 1 + 1.24e5iT - 3.93e9T^{2} \)
89 \( 1 + 1.87e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.21e5iT - 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.85775141697761078046822034470, −10.78248925728676171489123208182, −10.10398401248932038977442987821, −8.929658241469013755246467430626, −7.72903140963740348329713806726, −6.83127252628284953957525675402, −6.25950221181784898647210506887, −4.59751833812579403199625802151, −3.59282616789083288788442796944, −1.92313619288615407870611353433, 0.17675259521205686623124730068, 1.37641177058982749173768392286, 2.83180576882867425728023098930, 4.23610310322756103757916757032, 5.26884125820324736323026044839, 6.32503770363340673581227717120, 8.302896028928997588668931284761, 8.654915907508723626605928242699, 9.596359128307532474133060659246, 10.91261595397896166678584915753

Graph of the $Z$-function along the critical line